Differential forms on ringed spaces of valuation rings (Q1903378)

Let \(A\) be a subring of a field \(K\). The set of valuation rings of \(K\) containing \(A\) has a structure of local ringed space denoted by \(\text{Zar} (K|A)\) [cf. the author, Tokyo J. Math. 13, No. 2, 259-275 (1990; Zbl 0726.14001)]. The author constructs a category \({\mathcal C}_0 (K|A)\) of local ringed spaces containing both \(\text{Zar} (K|A)\) and all integral schemes proper over \(\text{Spec} A\) with rational function field \(K\). Furthermore for objects \(X\) of \({\mathcal C}_0 (K|A)\) divisor groups and sheaves \(\Omega^m_X\) of differential forms are introduced. If \(A\) is a perfect field and \(X\) a regular scheme then \(\Omega^m_X\) coincides with the ordinary sheaf of differential forms.